# Deriving a thermodynamics relationship

1. Sep 6, 2008

### Jacobpm64

1. The problem statement, all variables and given/known data
Derive:
$$\left(\frac{\partial C_{P}}{\partial P}\right)_{T} = -T \left(\frac{\partial^2 V}{\partial T^2}\right)_{P}$$

2. Relevant equations
I have already derived that
$$C_{P} = C_{V} + \left[\left(\frac{\partial E}{\partial V}\right)_{T} + P \right] \left( {\frac{\partial V}{\partial T}\right)_{P}$$

I also know that
$$C_{V} = \left( \frac{\partial E}{\partial T}\right)_{V}$$

I also know that you can write differentials like this:
$$z = z(x,y)$$
$$dz = \left( \frac{\partial z}{\partial x}\right)_{y} dx + \left( \frac{\partial z}{\partial y}\right)_{x} dy$$

3. The attempt at a solution
My problem is not knowing what variables $$C_{P}$$ is a function of. Therefore, I do not know how to write its differential (if you even can). I also do not know how to differentiate the above expression for $$C_{P}$$ with respect to P holding T constant.

I also do not know how to use Euler's chain rule for something like $$C_{P}$$ because i've never messed with anything that had more than 2 independent variables.

I have tried three pages worth of manipulating, but I do not know if i'm even manipulating correctly because of what i mentioned above.

Anything to get me started or tell me what is legal.. would be greatly appreciated..

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